The proof is an adaptation from [1]}, which is stated for Lebesgue absolutely continuous measures over \(\mathcal {T}=\mathbb {R}^d\) . In short, the linear dependence of the birth flux on the mass of \(\nu \)  gives a bound on this mass uniform in time, in which case both \(\kappa ^{\pm }_{\nu }\)  are Lipschitz in \(\nu \)  on \((\Gamma ,\Vert \cdot \Vert )\) , and classical existence theory can be applied.
